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Modulus of continuity of the Dirichlet solutions. (English) Zbl 1203.31008

Let \(D\) be a bounded domain in \(\mathbb{R}^{n}\) \((n\geq 2)\). For a function \(f:\partial D\rightarrow \mathbb{R}\) we denote by \(H^{D}f\) the corresponding solution to the Dirichlet problem in \(D\). This paper investigates when strong continuity properties of \(f\) are inherited by \(H^{D}f\). In an earlier paper [Bull. Lond. Math. Soc. 34, No. 6, 691–702 (2002; Zbl 1036.31003)] the author characterized those domains for which \(\beta \)-Hölder continuity (\(0<\beta <1\)) of \(f\) is inherited by \(H^{D}f\). (See also A. Hinkkanen [J. Anal. Math. 51, 1–29 (1988; Zbl 0672.31001)]). The present paper deals with this question for a general modulus of continuity. It is shown that the problem reduces to examining the Dirichlet solution of a natural test function, and this is then related to a decay property of harmonic measure. Contrary to what one might expect, smoothness of the boundary is not required. For example, it is shown that the weak modulus of continuity \((-\log t)^{-\alpha }\) \((\alpha >0)\) transfers from \(f\) to \(H^{D}f\) if \(D\) is a Lipschitz domain \((n\geq 2)\) or a finitely connected plane domain without punctures.

MSC:

31B25 Boundary behavior of harmonic functions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
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